A General Framework for Equivalences in Answer-Set Programming by Countermodels in the Logic of Here-and-There

Different notions of equivalence, such as the prominent notions of strong and uniform equivalence, have been studied in Answer-Set Programming, mainly for the purpose of identifying programs that can serve as substitutes without altering the semantics, for instance in program optimization. Such semantic comparisons are usually characterized by various selections of models in the logic of Here-and-There (HT). For uniform equivalence however, correct characterizations in terms of HT-models can only be obtained for finite theories, respectively programs. In this article, we show that a selection of countermodels in HT captures uniform equivalence also for infinite theories. This result is turned into coherent characterizations of the different notions of equivalence by countermodels, as well as by a mixture of HT-models and countermodels (so-called equivalence interpretations). Moreover, we generalize the so-called notion of relativized hyperequivalence for programs to propositional theories, and apply the same methodology in order to obtain a semantic characterization which is amenable to infinite settings. This allows for a lifting of the results to first-order theories under a very general semantics given in terms of a quantified version of HT. We thus obtain a general framework for the study of various notions of equivalence for theories under answer-set semantics. Moreover, we prove an expedient property that allows for a simplified treatment of extended signatures, and provide further results for non-ground logic programs. In particular, uniform equivalence coincides under open and ordinary answer-set semantics, and for finite non-ground programs under these semantics, also the usual characterization of uniform equivalence in terms of maximal and total HT-models of the grounding is correct, even for infinite domains, when corresponding ground programs are infinite.